ordinarily, the base of the logarithms is 10; that is, the logarithm is theexponent that would be affixed to 10 to produce the number corresponding to the logarithm. To illustrate, by taking simple numbers:
Logarithm of 10 = 1 because 101 = 10 Logarithm of 100 = 2 because 102 = 100 Logarithm of 1000 = 3 because 103 = 1000
In each case, it will be seen that the exponent of 10 equals the logarithm of the number. The logarithms of all numbers between 10 and 100 equal 1 plus some fraction. For example: The loga- rithm of 20 = 1.301030.
The logarithms of all numbers between 100 and 1000 = 2 plus some fraction; between 1000 and 10,000 = 3 plus some fraction;
and so on. The tables of logarithms in engineering handbooks give only this fractional part of a logarithm, which is called the man- tissa. The whole number part of a logarithm, which is called the characteristic, is not given in the tables because it can easily be determined by simple rules. The logarithm of 350 is 2.544068. The whole number 2 is the characteristic (see Handbook page 121) and the decimal part 0.544068, or the mantissa, is found in the table (Handbookpage 125).
Principles Governing the Application of Logarithms.—W h e n logarithms are used, the product of two numbers can be obtained as follows: Add the logarithms of the two numbers; the sum equals the logarithm of the product. For example: The logarithm of 10 (commonly abbreviated log 10) equals 1; log 100 = 2; 2 + 1 = 3, which is the logarithm of 1000 or the product of 100 ì 10.
Logarithms would not be used for such a simple example of multiplication; these particular numbers are employed merely to illustrate the principle involved.
For division by logarithms, subtract the logarithm of the divisor from the logarithm of the dividend to obtain the logarithm of the quotient. To use another simple example, divide 1000 by 100 using logarithms. As the respective logarithms of these numbers are 3 and 2, the difference of equals the logarithm of the quotient 10.
LOGARITHMS 34
In using logarithms to raise a number to any power, simply mul- tiply the logarithm of the number by the exponent of the number;
the product equals the logarithm of the power. To illustrate, find the value of 103 using logarithms. The logarithm of 10 = 1 and the exponent is 3; hence, 3 ì 1 = 3 = log of 1000; hence, 103 = 1000.
To extract any root of a number, merely divide the logarithm of this number by the index of the root; the quotient is the logarithm of the root. Thus, to obtain the cube root of 1000 divide 3 (log 1000) by 3 (index of root); the quotient equals 1 which is the loga- rithm of 10. Therefore,
Logarithms are of great value in many engineering and shop calculations because they make it possible to solve readily cumber- some and also difficult problems that otherwise would require complicated formulas or higher mathematics. Keep constantly in mind that logarithms are merely exponents. Any number might be the base of a system of logarithms. Thus, if 2 were selected as a base, then the logarithm of 256 would equal 8 because 28 = 256.
However, unless otherwise mentioned, the term “logarithm” is used to apply to the common or Briggs system, which has 10 for a base.
The tables of common logarithms are found on Handbook pages 125 and 126. The natural logarithms, pages 127 and 128, are based upon the number 2.71828. These logarithms are used in higher mathematics and also in connection with the formula to determine the mean effective pressure of steam in engine cylin- ders.
Finding the Logarithms of Numbers.—There is nothing compli- cated about the use of logarithms, but a little practice is required to locate readily the logarithm of a given number or to reverse this process and find the number corresponding to a given logarithm.
These corresponding numbers are sometimes called “antiloga- rithms.”
Study carefully the rules for finding logarithms given on Hand- bookpages 121 to 124 Although the characteristic or whole-num- ber part of a logarithm is easily determined, the following table will assist the beginner in memorizing the rules.
31000 = 10
LOGARITHMS 35
Example of the use of the table of numbers and their character- istics: What number corresponds to the log 2.55145? Find 0.551450 in the log tables to correspond to 356. From the table of characteristics, note that a 2 characteristic calls for one zero in front of the first integer; hence, point off 0.0356 as the number cor- responding to the log 2.55145. Evaluating logarithms with nega- tive characteristics is explained more thoroughly later.
Example 1:Find the logarithm of 46.8.
The mantissa of this number is 0.670246. When there are two whole-number places, the characteristic is 1; hence, the log of 46.8 is 1.670246.
After a little practice with the above table, one becomes familiar with the rules governing the characteristic so that reference to the table is no longer necessary.
Obtaining More Accurate Values Than Given Directly by Tables.—The method of using the tables of logarithms to obtain more accurate values than are given directly, by means of interpo- lation, is explained on Handbook page 122. These instructions should be read carefully in order to understand the procedure in connection with the following example:
Example 2:
Sample Numbers and Their Characteristics Characteristic Number Characteristic Number
0.008 3 88 1
0.08 2 888 2
0.8 1 8888 3
8.0 0 88888 4
log 76824 = 4.88549 log numerator = 6.60213 log 52.076 = 1.71664 − log 435.21 = 2.63870 log numerator = 6.60213 log quotient = 3.96343
76824ì52.076 435.21 --- =
LOGARITHMS 36
The number corresponding to the logarithm 3.96343 is 9192.4.
The logarithms just given for the dividend and divisor are obtained by interpolation in the following manner:
In the log tables on page 126 of the Handbook, find the man- tissa corresponding to the first three digits of the number 76824, and the mantissa of the next higher 3-digit number in the table, 769. The mantissa of 76824 is the mantissa of 768 plus 24⁄100 times the difference between the mantissas of 769 and 768.
Thus, log 76824 = 0.24 ì 0.000565 + log 76800 = 4.885497.
The characteristic 4 is obtained as previously illustrated in the table on page 35. By again using interpolation as explained in the Handbook, the corrected mantissas are found for the logarithms of 52.076 and 435.21.
After obtaining the logarithm of the quotient, which is 3.96343, interpolation is again used to determine the corresponding number more accurately than would be possible otherwise. The mantissa .96343 (see Handbook page 126) is found, in the table, between 0.963316 and 0.963788, the mantissas corresponding to 919 and 920, respectively.
Note that the first line gives the difference between the two mantissas nearest .96343, and the second line gives the difference between the mantissa of the quotient and the nearest smaller man- tissa in the Handbook table. The characteristic 3 in the quotient 3.96343 indicates 4 digits before the decimal point in the answer, thus the number sought is 9190 + 114⁄472(9200−9190) = 9192.4.
Changing Form of Logarithm Having Negative Characteris- tic.—The characteristic is frequently rearranged for easier manip- ulation. Note that 8 − 8 is the same as 0; hence, the log of 4.56 could be stated: 0.658965 or 8.658965 − 8. Similarly, the log of 0.075 = 2.875061 or 8.875061 − 10 or 7.875061 − 9. Any similar
Mantissa 769 = .885926 Mantissa 768 = .885361 Difference = .000565
0.963788− 0.963316 = 0.000472 0.96343− 0.963316 = 0.000114
LOGARITHMS 37